If you're going to do a "one per answer" type question, please make it community wiki (click "edit" and check the "community wiki" box in the lower right).
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Anton GeraschenkoOct 22 '09 at 5:01

Do you want coarse moduli spaces that are not fine? Do you want the examples to be algebraic? Most of my favorite spaces are moduli spaces.
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Tom ChurchOct 22 '09 at 5:29

To the first question: yes, the ideal example would be one where the coarse moduli space is very far from the stack, and difficult to guess. (At present, I can only guess the coarse moduli space for stacks of the form [X/G], and know the answers for A_g, M_g.) To the second question: non-algebraic examples are cool.
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Thanos D. PapaïoannouOct 22 '09 at 5:44

6 Answers
6

Here * is a point and G is an algebraic group. Its sections on a scheme X form the groupoid of principal G-bundles on X. Because principal G bundles are locally trivial, the coarse moduli space is a point.

If M is a stack, it determines a presheaf \pi_0(M), whose sections are the isomorphisms classes of objects of M. One way to think about the coarse moduli space is as a space representing the sheafification of this presheaf in whatever topology you are working with.

Starting from this example, you should also be able to work out the issues surrounding [X/G]. In fact, it's good to revisit BG after working out the general case. But I won't do it here since you asked for one example only. In fact, maybe I should have said

*= Spec(Q)

and

G=GL_2/Q.

I learned about stacks in the days before all these nice books, so I'm not sure about a reference for anything I write about them. But the statements here should all be pretty clear from the definitions.

Personal favorite: The Jacobian J of a smooth curve, which is the coarse moduli space for degree zero line bundles on that curve. If you choose a point in your curve, you can also realize the Jacobian as the stack which classifies pairs (L,t) where L is a degree zero line bundle and t is a point in the fiber of L over your chosen point (i.e. a trivialization). There's an obvious surjective map from this stack to the stack M of degree zero line bundles; just forget the trivialization. Thus, the Jacobian is both a coarse moduli space for M and an atlas for M.

Whenever I'm doing something and I'm trying to see what difference coarse and fine moduli spaces make, I test the theorem with the STACK M_1 of genus 1 curves and with the variety which is the coarse moduli space of genus 1 curves, A^1, so A^1 is one of my favorite coarse moduli spaces.

Thank you all for your very kind answers! It was silly of mine to suggest we rate the examples, since it's unclear if any examples are better than others. To atone for my mistake, let me offer a summary of the proposed examples in the form of a table.

stack : coarse moduli space

$\mathcal{M}_1$ : affine line $\mathbf{A}^1$;

line bundles of degree $0$ on a smooth curve : $\mathrm{Pic}^0$;

$[\mathrm{pt}/G]$ : $\mathrm{pt}$.

Good moduli spaces

One thing that I learned from Alper, Good moduli spaces is that an object technically better than the coarse moduli space is a good moduli space, a replacement notion which commutes with arbitrary, not just flat, base change, and exists more generally. Explicitly, a good moduli space of an Artin stack $X$ is a morphism $f$ to an algebraic space such that 0) it's quasi-compact a) pushforward along $f$ is exact on quasi-coherent sheaves, and b) the pullback morphism $f_*\mathcal{O}_Y\rightarrow \mathcal{O}_X$ is an isomorphism.

For example, given a linear algebraic group $G$ acting an an affine scheme of ring A over a field, the morphism
$$
[\mathrm{Spec}(A)/G] \rightarrow \mathrm{Spec}(A^G)
$$
is a good moduli space (ibid., Example 8.3), hence over $\mathbf{Q}$,
$$
[\mathbf{A}^1/\mathrm{Spec}(k)]\rightarrow \mathrm{Spec}(k)
$$
is a good moduli space (ibid., Example 8.1 and Example 12.4 (1)).

And when is the old notion of a coarse moduli space a special case of the new notion of good moduli space, you ask? Well, if the stack in question is tame (ibid. Example 8.1), i.e., if it has inertia stack finite over it and if the morphism from it to the coarse moduli space is exact on quasicoherent sheaves. (The latter condition is automatic in characteristic $0$.)